Beam relationships: sign change in shear force and its effect on bending moment Statement: If the shear force at a point changes sign from positive to negative (or vice versa), then the bending moment at that point is zero.

Introduction / Context:
Shear force V and bending moment M are linked through calculus: dM/dx = V. Understanding this relationship is critical for drawing correct shear force and bending moment diagrams and identifying locations of maximum or minimum bending moment.

Given Data / Assumptions:

• Beam subject to transverse loading.
• V changes sign at some section x = x0.
• Linear elastic, small deflection theory.

Concept / Approach:
Because dM/dx = V, when V = 0 at a point, M has a stationary value (local maximum or minimum), not necessarily zero magnitude. The magnitude of M depends on boundary conditions and loading history up to that point. Zero bending moment occurs at free ends of simple beams or at internal hinges, not generically where V crosses zero.

Step-by-Step Solution:
At x0, V(x0) = 0 ⇒ dM/dx|_{x0} = 0.A zero derivative implies extremum of M, not M = 0.Therefore, the statement asserting M = 0 whenever V changes sign is false.

Verification / Alternative check:
For a simply supported beam with uniform load, V = 0 at mid-span but M is maximum, not zero. This counterexample confirms the statement is false.

Why Other Options Are Wrong:
The conditional True variants (cantilever, UDL, fixed supports) remain incorrect; the calculus relationship does not change for those cases.

Common Pitfalls:
Equating stationary bending moment with zero value; misapplying support boundary conditions.

Final Answer:
False